Assigned work: pg.407 #1-13

Recall dot product produces a scalar from two vectors.

Today we examine a Cross Product (or Vector Product) which produces a vector from two vectors.

7.6 Cross Product

Cross Product of will be a vector that is perpendicular to both . Therefore the cross product is ONLY defined in R3.

It is useful in physical problems such as torque and area of a parallelogram (applications we will discuss tomorrow)

a and b

a and b

7.6 Cross Product

Prove Cross Product formula :

7.6 Cross Product

Cross Product of

is the vector:

1 2 3 1 2 3( , , ) ( , , )a a a a and b b b b

2 3 3 2 3 1 1 3 1 2 2 1( , , )a b a b a b a b a b a b a b

7.6 Cross Product

An easier method to remember the Cross Product formula is:

2 3 3 1 1 2

2 3 3 1 1 2

, ,a a a a a a

a bb b b b b b

where

a bad bc

c d

7.6 Cross Product

Finding a Vector Perpendicular to Two Vectors:

If are two non-collinear vectors in 3D, then every vector perpendicular to bothis of the form where

a and b

a and b

( )k a b

k R

7.6 Cross Product

Ex 1: a)Find a vector perpendicular to the vectors (2,5,0) and (-4,0,9).

Answer: (45,-18,20)

b)Check your answer using Dot Product (since dot product of 2 perpendicular vectors should be 0).

7.6 Cross Product

Magnitude of the Cross Product of is:

where is the angle between and

a and b

sina b a b

a and b

0 180

7.6 Cross Product

form what we call a right handed system……

Place your right hand on the diagram so that your finger curled in the direction from is an angle less than . The direction of will be the direction your thumb points.

Direction of Cross Product of is such that:

, ,a b and a b

a b

a to b

180 a b

7.6 Cross Product

Direction of Cross Product of :

Thumb is in so: Thumb is out so:

is into page is out of page

a b

b

b

a

a

Fingers curl this way

Fingers curl this way

a b

a b

7.6 Cross Product

Direction of Cross Product of :

Note there are other methods of this right hand rule to find the direction. Some of you may be used to using the first vector as the thumb, the second vector as your fingers and the direction of the cross product as the palm of your hand. (Motor right hand rule in Grade 11 Physics)

Either method works – use which one you like the best.

a b

7.6 Cross Product

Ex 2:If and the angle betweenis 45 degrees Determine the magnitude and direction of . (include a diagram).

Answer: Magnitude is and direction will depend on how you drew your diagram.

5 6a and b

a and b

a b

15 2

7.6 Cross Product

Ex 3:Find the cross product of :

Answer:

33 1 18i j k

3 2 4 6 7a i j k and b i j k

7.6 Cross Product

Properties of Cross Product:

Let be vectors in 3D

1) Order matters (anti-commutative)

2) Distributive Law

3)

, ,a b and c

a b b a

a b c a b a c

k a b ka b a kb

7.6 Cross Product

Ex 4:Given:Determine:

(Note: Think about what order it should be done. Cross product MUST go 1st since you cannot do cross product of a vector and a scalar)

Answer: 78

(6, 2, 3) (5,1, 4) (4,2,1)a b c

a b c

7.6 Cross Product

Important READ 7.6 before doing the assigned questions.

Make a note on:1)What is a “Triple Scalar Product”?2)What is a Triple Vector Product”?